Modern Physics Letters B, Vol. 23, No. 26 (2009) 3147–3155 c  World Scientific Publishing Company NEW ANALYTICAL SOLUTION OF THE THREE-DIMENSIONAL NAVIER STOKES EQUATIONS MOHAMMAD MEHDI RASHIDI *, ‡ and GANJI DOMAIRRY † * Engineering Faculty of Bu-Ali Sina University, P. O. Box 65175-4161, Hamedan, Iran † Mechanical Engineering Department, Mazandaran University, Babol, P. O. Box 484, Iran ‡ mm rashidi@yahoo.com Received 24 January 2009 Revised 15 March 2009 The purpose of this study is to implement a new analytical method (the DTM-Pad´ e technique, which is a combination of the differential transform method (DTM) and the Pad´ e approximation) for solving Navier–Stokes equations. In this letter, we will consider the DTM, the homotopy perturbation method (HPM) and the Pad´ e approximant for finding analytical solutions of the three-dimensional viscous flow near an infinite rotat- ing disk. The solutions are compared with the numerical (fourth-order Runge–Kutta) solution. The results illustrate that the application of the Pad´ e approximants in the DTM and HPM is an appropriate method in solving the Navier–Stokes equations with the boundary conditions at infinity. On the other hand, the convergence of the obtained series from DTM-Pad´ e is greater than HPM-Pad´ e. Keywords : DTM-Pad´ e; HPM-Pad´ e; Navier–Stokes equations. 1. Introduction The HPM was proposed first by He in 1998. This method was further developed and improved by He and applied to nonlinear oscillators with discontinuities, 1 nonlinear wave equations, 2 asymptotology, 3 boundary value problem, 4 limit cycle and bifur- cation of nonlinear problems 5 and many other subjects. 6 In this paper, we will show that the HPM and the DTM are not applicable for solving the three-dimensional Navier–Stokes equations with the boundary conditions at the infinity. By using the Pad´ e approximant, 7,8 we can solve this problem with the HPM and DTM. The concept of the DTM was first proposed by Zhou, 9 who solved linear and nonlinear problems in electrical circuit problems. Chen and Ho 10 developed this method for partial differential equations and Ayaz 11 applied it to the system of differential equations The motivation of this letter is to extend HPM-Pad´ e and DTM-Pad´ e to solve the three-dimensional Navier–Stokes equations. ‡ Corresponding author. 3147